Bearing estimation is an important topic in sonar and radar applications. Historically, bearing estimation has been performed by a process called beamforming. The conventional beamformer adds delays to the outputs of sensors along an array. The summation of these signals produces a beam steered in a direction determined by the delay intervals. The angular resolution of this method is given by the wavelength to aperture ratio. This is known as the Rayleigh limit.
Many techniques have been used to increase the resolution of the bearing estimation problem. The most promising technique is the maximum likelihood method. This technique has been shown to approach the theoretical limit of resolution (Cramer-Rao bound). It is a robust method that works well even with low signal to noise ratios.
The maximum likelihood method fits the data to a model. Typical applications use a least squares metric to calculate the likelihood function. A search based on finding the most likely model parameters results in the best bearing estimates. Mathematically, the likelihood function can be written as: ##EQU1## where L=likelihood function
t=time PA1 M=number of snapshots PA1 y=observation vector (length=p) PA1 s=signal vector (length=q) PA1 D=steering matrix PA1 .omega.=narrowband frequency PA1 .tau.(.theta.)=time delay between elements
For a uniform linear array, the steering matrix has the following Vandermonde structure: ##EQU2## where K.sub.i =.omega..tau.(.theta..sub.i)
Minimization of the likelihood function with respect to the signal vector, s, can be accomplished by setting the partial derivatives of L to zero. This yields the following relationship EQU s(t)=(D.sup.H D).sup.-1 D.sup.H y(t) (2)
where H denotes the Hermitian transposition. Inserting this result into the likelihood equation yields ##EQU3## Defining the projection operator EQU p=I-D(D.sup.H D).sup.-1 D.sup.H ( 4)
yields ##EQU4## Equation (5) may be rewritten as EQU L=tr[PR] (6)
where R, from equation (6) is the sample covariance matrix ##EQU5## The representation of equation (6) is the textbook approach to the maximum likelihood method.
The maximum likelihood method can also be applied to spectral estimation problems. Mathematically, the equation differences include replacing the wavenumber, k, with the angular frequency variable, .omega.. Likewise the spatially separated array data is replaced with time series data. This duality between the spatial domain and the time domain is seen in many areas of signal processing.
The utility of the maximum likelihood method has been limited because of its large computational load. The likelihood function evaluation has traditionally required that the complex matrix operations to be done numerically. The computational load of the matrix computations scale as the number of array elements squared. This quadratic increase in the processing load requirement tends to limit the usefulness of this approach.
The goal of the maximum likelihood method is, of course, to find the largest value of the likelihood function. This is a nonlinear multivariate problem. These searches can be difficult. An additional complication occurs when the number of sources is not known. This affects the number of variables and therefore the size of the steering matrix. The solution to this overall maximum likelihood problem is computationally difficult.